Question

A stock price is currently $20. It is known that at the end of one month that the stock price will either increase to 22 or decrease to 16. The risk-free interest rate is 12% per annum with continuous compounding. The hedge portfolio is a long position in Δ shares of stock plus one short Euorpean call option with strike price of $20 and expiration in 1 month. Using the no-arbitrage method, what is the present value of this hedge portfolio at time 0?

Answer 1

K = $ 20

Payoff in the up state i.e. when stock price is = S_u = $ 22,

Payoff from hedge portfolio = S_uΔ - max (S_u - K, 0) = 22Δ - max (22 - 20, 0) = 22Δ - 2

Payoff in the down state i.e. when stock price is = S_d = $ 16,

Payoff from hedge portfolio = S_dΔ - max (S_d - K, 0) = 16Δ - max (16 - 20, 0) = 16Δ

Since it's a hedge portfolio, payoff from the portfolio in either state should be the same.

Hence, 22Δ - 2 = 16Δ

Hence, Δ = 2 / 6 = 1/3.

And the value of the hedge portfolio = 22Δ - 2 = 16Δ = 16 x 1/3 = 16 / 3

Since the portfolio is a hedge portfolio, it must earn the risk-free rate of interest. Hence, the present value of this hedge portfolio = PV = FV x e^-rt = 16 / 3 x e^{-12% x 1/12} = $  5.28

Hence, the final answer is $ 5.28